After catching my starter Pokémon I've not encountered any more Pokémon, is it entirely random when Pokémon pop up or is there some kind of indicator for me to be on the lookout for?
Pokemon spawn more often in places that have lots of people (based on Google map data). This means places like malls have more Pokemon than fields or forests. If there are no Pokemon near you, then try visiting a popular shopping center, gas stations, a busy intersection, or any other place where people are likely to use Google maps. You will know that you have found a place where Pokemon spawn when you see a leaf animation. Note that this does not necessarily mean a Pokemon is currently there, only that they regularly spawn there, so it's a good place to look.
You can also use incense to have a Pokemon spawn on your location every 5 minutes for 30 minutes, regardless of your location, but that will require real money to do regularly.